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We propose an implementation of symplectic implicit Runge-Kutta schemes forhighly accurate numerical integration of non-stiff Hamiltonian systems based onfixed point iteration. Provided that the computations are done in a givenfloating point arithmetic, the precision of the results is limited by round-offerror propagation. We claim that our implementation with fixed point iterationis near-optimal with respect to round-off error propagation under theassumption that the function that evaluates the right-hand side of thedifferential equations is implemented with machine numbers (of the prescribedfloating point arithmetic) as input and output. In addition, we present asimple procedure to estimate the round-off error propagation by means of aslightly less precise second numerical integration. Some numerical experimentsare reported to illustrate the round-off error propagation properties of theproposed implementation.

Reducing and monitoring round-off error propagation for symplectic implicit Runge-Kutta schemes